Problem: What is the extraneous solution to these equations? $\dfrac{x^2 + 14}{x - 7} = \dfrac{x + 86}{x - 7}$
Explanation: Multiply both sides by $x - 7$ $ \dfrac{x^2 + 14}{x - 7} (x - 7) = \dfrac{x + 86}{x - 7} (x - 7)$ $ x^2 + 14 = x + 86$ Subtract $x + 86$ from both sides: $ x^2 + 14 - (x + 86) = x + 86 - (x + 86)$ $ x^2 + 14 - x - 86 = 0$ $ x^2 - 72 - x = 0$ Factor the expression: $ (x + 8)(x - 9) = 0$ Therefore $x = -8$ or $x = 9$ The original expression is defined at $x = -8$ and $x = 9$, so there are no extraneous solutions.